Problem: A circle has a sector with area $40\pi$ and central angle $\dfrac{4}{5}\pi$ radian. What is the area of the circle? ${100\pi}$ $\color{#9D38BD}{\dfrac{4}{5}\pi}$ ${40\pi}$
The ratio between the sector's central angle $\theta$ and $2 \pi$ radians is equal to the ratio between the sector's area, $A_s$ , and the whole circle's area, $A_c$ $\dfrac{\theta}{2 \pi} = \dfrac{A_s}{A_c}$ $\dfrac{4}{5}\pi \div 2 \pi = 40\pi \div A_c$ $\dfrac{2}{5} = 40\pi \div A_c$ $A_c \times \dfrac{2}{5} = 40\pi$ $A_c = 40\pi \times \dfrac{5}{2}$ $A_c = 100\pi$